<p>This paper is dedicated to studying matrix solutions of the cubic Szegő equation on the real line, which is introduced in Pocovnicu [Anal PDE 4(3):379–404, 2011; Dyn Syst A 31(3):607–649, 2011] and Gérard–Pushnitski (Commun Math Phys 405:167, 2024), leading to the following cubic matrix Szegő equation on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11040_2025_9500_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>, <Equation ID="Equ102"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11040_2025_9500_Article_Equ102.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="401" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} i \partial _t U = \Pi _{\ge 0} \left( U U ^* U \right) , \quad \widehat{\left( \Pi _{\ge 0} U\right) }(\xi )= {\textbf{1}}_{\xi \ge 0}{\hat{U}}(\xi )\in {\mathbb {C}}^{M \times N}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>i</mi> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>U</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mfenced close=")" open="("> <mi>U</mi> <msup> <mi>U</mi> <mo>∗</mo> </msup> <mi>U</mi> </mfenced> <mo>,</mo> <mspace width="1em" /> <mover accent="true"> <mfenced close=")" open="("> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mi>U</mi> </mfenced> <mo stretchy="true">^</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mn mathvariant="bold">1</mn> <mrow> <mi>ξ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mover accent="true"> <mi>U</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mi>M</mi> <mo>×</mo> <mi>N</mi> </mrow> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Inspired by the space-periodic case in Sun (The matrix Szegő equation, <a href="http://arxiv.org/abs/2309.12136">arXiv:2309.12136</a>), we establish its Lax pair structure via double Hankel operators and Toeplitz operators. Then the explicit formula in Gérard–Pushnitski (Commun Math Phys 405:167, 2024) can be extended to two equivalent formulas in the matrix equation case, which both express every solution explicitly in terms of its initial datum and the time variable.</p>

Matrix Solutions of the Cubic Szegő Equation on the Real Line

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发表日期：

2025年3月9日

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Ruoci Sun

摘要

This paper is dedicated to studying matrix solutions of the cubic Szegő equation on the real line, which is introduced in Pocovnicu [Anal PDE 4(3):379–404, 2011; Dyn Syst A 31(3):607–649, 2011] and Gérard–Pushnitski (Commun Math Phys 405:167, 2024), leading to the following cubic matrix Szegő equation on ${\mathbb {R}}$ $R$ , $\begin{aligned} i \partial _t U = \Pi _{\ge 0} \left( U U ^* U \right) , \quad \widehat{\left( \Pi _{\ge 0} U\right) }(\xi )= {\textbf{1}}_{\xi \ge 0}{\hat{U}}(\xi )\in {\mathbb {C}}^{M \times N}. \end{aligned}$ $\begin{matrix} i \partial_{t} U = Π_{\geq 0} (U, U^{*}, U), \hat{(Π_{\geq 0}, U)} (ξ) = 1_{ξ \geq 0} \hat{U} (ξ) \in C^{M \times N} . \end{matrix}$ Inspired by the space-periodic case in Sun (The matrix Szegő equation, arXiv:2309.12136), we establish its Lax pair structure via double Hankel operators and Toeplitz operators. Then the explicit formula in Gérard–Pushnitski (Commun Math Phys 405:167, 2024) can be extended to two equivalent formulas in the matrix equation case, which both express every solution explicitly in terms of its initial datum and the time variable.